TG EAPCET 2025 (Online) 4th May Evening Shift | TS EAMCET Year Wise Previous Years Questions

Chemistry

1. Observe the following statements. Statement -I Rutherford model of an atom cannot explain the stability of an atom. Stat 2. In hydrogen atom, an electron is transferred from an orbit of radius 1.3225 nm to another orbit of radius 0.2116 nm . Wh 3. Identify the correct orders regarding atomic radii (i) $\mathrm{Cl}>\mathrm{F}>\mathrm{Li}$ (ii) $\mathrm{P}>\mathrm{C}> 4. $$ \text { Match the following } $$ $$ \begin{array}{cccc} \hline & \text { List-I (Elements) } & & \text { List-II (Gro 5. The atomic numbers of the elements $X, Y, Z$ are $a, a+1, a+2$ respectively. $Z$ is an alkali metal. The nature of bondi 6. The sets of molecules in which central atom has no lone pair of electrons are I. $\mathrm{SnCl}_2, \mathrm{NH}_3, \mathr 7. The isobars of one mole of an ideal gas were obtained at three different pressure ( $p_1, p_2$ and $p_3$ ). The slopes o 8. 100 mL of $0.05 \mathrm{M} \mathrm{Cu}^{2+}$ aqueous solution is added to IL of 0.1 M KI solution. The number of moles o 9. The $C_p$ of an ideal gas is $10314 \mathrm{Jmol}^{-1} \mathrm{~K}^{-1}$. One mole of this gas is expanded against a con 10. At $T(\mathrm{~K}), K_p$ value for the reaction, $$ 2 \mathrm{AO}_2(\mathrm{~g})+\mathrm{O}_2(\mathrm{~g}) \rightlefthar 11. A sample of water contains $\mathrm{Mg}\left(\mathrm{HCO}_3\right)_2$ and $\mathrm{Ca}\left(\mathrm{HCO}_3\right)_2$ On 12. Which of the following statements is not correct? 13. The order of negative standard potential values of Li, $\mathrm{Na}, \mathrm{K}$ is 14. In which of the following reactions, hydrogen is evolved? I. Reaction of sodium borohydride with iodine II. Oxidation of 15. Which of the following statements is not correct regarding the gas evolved by the reaction of dilute HCl on $\mathrm{CaC 16. Observe the following statements Statement I The carbon containing components of photochemical smog are acrolein, methan 17. The condensed, bond line and complete formulae of $n$-butane are respectively. 18. ' $X$ ' is the isomer of $\mathrm{C}_6 \mathrm{H}_{14}$. It has four primary carbons and two tertiary carbons, ' $X$ ' c 19. $$ \text { What are } X \text { and } Y \text { in the following reaction sequence? } $$ $$ \text { Isopentane } \xright 20. What are $X$ and $Y$ respectively in the following reactions? 21. A metal ( $M$ ), crystallises in fcc lattice with edge length of $4.242 \mathop {\rm{A}}\limits^{\rm{o}}$. What is the r 22. A solid mixture weighing 5 g contains equal number of moles of $\mathrm{Na}_2 \mathrm{CO}_3$ and NaHCO . This solid mixt 23. At 298 K the equilibrium constant for the reaction $M(s)+2 \mathrm{Ag}^{+}(a q) \longrightarrow M^{2+}(a q)+2 \mathrm{Ag 24. $A \rightarrow P$ is a first order reaction. At 300 K this reaction was started with $[A]=0.5 \mathrm{~mol} \mathrm{~L}^ 25. Observe the following reaction I. $\mathrm{CO}(\mathrm{g})+\mathrm{H}_2(\mathrm{~g}) \xrightarrow{X} \mathrm{HCHO}(\math 26. Composition of siderite ore is 27. Which of the following gives more number of oxides on reacting with HCl ? 28. The number of lone pairs of electrons on the central atom of $\mathrm{XeO}_3, \mathrm{XeOF}_4$ and $\mathrm{XeF}_6$ resp 29. Which of the following statements is not correct? 30. The pair of ions with paramagnetic nature and same number of electrons is 31. $$ \text { Observe the following complex ions } $$ $$ \begin{array}{cccc} \hline\left[\mathrm{Mn}(\mathrm{CN})_6\right]^ 32. The polymer chains are held together by hydrogen bonding in a polymer $X$. Polymer $X$ is formed from monomers $Y$ and $ 33. Amino acid ' $X$ ' contains phenolic hydroxy group and amino acid ' $Y$ ' contains amide group. ' $X$ ' and ' $Y$ ' resp 34. The chemical $X$ is used in the prevention of heart attack. The structure of $X$ is 35. $$ \text { Observe the following reactions } $$ The correct order of reactivity of $X, Y, Z$ towards $\mathrm{S}_{\math 36. An alochol $X\left(\mathrm{C}_5 \mathrm{H}_{12} \mathrm{O}\right)$ produces turbidity instantly with conc. $\mathrm{HCl} 37. Toluene on reaction with reagent $A$ gives $X$. This $(X)$ forms 2, 4 dinitrophenylhydrazone and reduces ammoniacal silv 38. $$ \begin{aligned} &\text { What are } X \text { and } Y \text { in the following reaction sequence? }\\ &\mathrm{C}_5 \ 39. $$ \text { Observe the following set of reactions } $$ $$ \text { What are } X, Y \text { and } Z \text { respectively 40. Consider the following set of reactions What are $A$ and $B$ respectively?

Mathematics

1. The domain and range of $f(x)=\frac{1}{\sqrt{|x|-x^2}}$ are $A$ and $B$ respectively. Then $A \cup B=$ 2. A function $f: R \rightarrow R$ defined by $$ f(x)=\left\{\begin{array}{c} 2 x+3, x \leq \frac{4}{3} \\ -3 x^2+8 x, x>\f 3. If $2^{4 n+3}+3^{3 n+1}$ is divisible by $P$ for all natural numbers $n$, then $P$ is 4. A is a $3 \times 3$ matrix satisfying $A^3-5 A^2+7 A+I=0$ If $A^5-6 A^4+12 A^3-6 A^2+2 A+2 I=l A+m I$, then $l+m=$ 5. If $A=\left[\begin{array}{lll}0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & x & 1\end{array}\right], A^{-1}=\frac{1}{2}\left[\begin{arra 6. Consider a homogeneous system of three linear equations in three unknowns represented by $A X=0$. If $X=\left[\begin{arr 7. The number of real values of ' $a$ ' for which the system of equations $2 x+3 y+a z=0, x+a y-2 z=0$ and $3 x+y+3 z=0$ ha 8. If the eight vertices of a regular octagon are given by the complex number $\frac{1}{x_j-2 i}(j=1,2,3,4,5,6,7,8)$, then 9. If $\left|Z_1-3-4 i\right|=5$ and $\left|Z_2\right|=15$, then the sum of the maximum and minimum values of $\left|Z_1-Z_ 10. If $Z=r(\cos \theta+i \sin \theta),\left(\theta \neq-\frac{\pi}{2}\right)$ is solution of $x^3=i$, then $r^9(\cos \theta 11. If $\omega \neq 1$ is a cube root of unity, then one root among the 7th roots of $(1+\omega)$ is 12. $f(x)=x^2-2(4 k-1) x+g(k)>0, \forall x \in R$ and for $k \in(a, b)$. If $g(k)=15 k^2-2 k-7$, then 13. If local maximum of $f(x)=\frac{a x+b}{(x-1)(x-4)}$ exists at $(2,-1)$, then $a+b=$ 14. If $1+2 i$ is a root of the equation $x^4-3 x^3+8 x^2-7 x+5=0$, then sum of the squares of the other roots is 15. If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+\frac{a}{2} x+b=0$ and $(\alpha-\beta)(\alpha-\gamma),(\be 16. All the letters of the word MOTHER are arranged in all possible ways and the resulting words (may or may not have meanin 17. The number of positive integral solution of $\frac{1}{x}+\frac{1}{y}=\frac{1}{2025}$ is 18. The number of positive integral solutions of $x y z=60$ is 19. Numerically greatest term in the expansion of $(3 x-4 y)^{23}$ when $x=\frac{1}{6}$ and $y=\frac{1}{8}$ is 20. Let $K$ be the number of rational terms in the expansion of $(\sqrt{2}+\sqrt[3]{3})^{6144}$. If the coefficient of $x^P( 21. If $\frac{3 x+1}{(x-1)^2\left(x^2+1\right)}=\frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C x+D}{x^2+1}$, then $2(A-C+B+D)=$ 22. If $\tan \left(\frac{\pi}{4}+\frac{\alpha}{2}\right)=\tan ^3\left(\frac{\pi}{4}+\frac{\beta}{2}\right)$, then $\frac{3+\ 23. If $P=\sin \frac{2 \pi}{7}+\sin \frac{4 \pi}{7}+\sin \frac{8 \pi}{7}$ and $Q=\cos \frac{2 \pi}{7}+\frac{4 \pi}{7}+\cos \ 24. If $\cos \alpha=\frac{l \cos \beta+m}{l+m \cos \beta}$, then $\left(\frac{\tan \frac{\alpha}{2}}{\tan \frac{\beta}{2}}\r 25. If $a, b$ are real numbers and $\alpha$ is a real roots of $x^2+12+3 \sin (a+b x)+6 x=0$, then the value of $\cos (a+b \ 26. The number of real solution of $\tan ^{-1} x+\tan ^{-1} 2 x=\frac{\pi}{4}$ is 27. Consider the following statements Statement $\mathrm{I} \cosh ^{-1} x=\tanh ^{-1} x$ has no solution Statement II $\cosh 28. If the angular bisector of the angle $A$ of the $\triangle A B C$ meets its circumcircle at $E$ and the opposite side $B 29. In a $\triangle A B C, a=5, b=4$ and $\tan \frac{C}{2}=\sqrt{\frac{7}{9}}$, then its inradius $r=$ 30. Two adjacent sides of a triangle are represented by the vectors $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ 31. A plane $\pi_1$ contains the vectors $\hat{\mathbf{i}}+\hat{\mathbf{j}}$ and $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$. Anot 32. In a quadrilateral $A B C D, \mathbf{A}=\frac{2 \pi}{3}$ and $A C$ is the bisector of angle $\mathbf{A}$. If $15|\mathbf 33. $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are three non- coplanar and mutually perpendicular vectors of same magnitude $K . r 34. Consider the following Assertion (A) The two lines $\mathbf{r}=\mathbf{a}+t(\mathbf{b})$ and $\mathbf{r}=\mathbf{b}+s(\m 35. The mean deviation about median of the numbers $3 x, 6 x, 9 x, \ldots .81 x$ is 91 , then $|x|=$ 36. Functions are formed from the set $A=\left\{a_1, a_2, a_3\right\}$ to another set $B=\left\{b_1, b_2, b_3, b_4, b_5\righ 37. $A$ and $B$ are two events of a random experiment such that $P(B)=0.4, P(A \cap \bar{B})=0.5, P(A \cup B)+P\left(\frac{B 38. There are two boxes each containing 10 balls. In each box, few of them are black balls and rest are white. A ball is dra 39. In a shelf there are three mathematics and two physics books. A student takes a book randomly. If he randomly takes, suc 40. In possion distribution, if $\frac{P(x=5)}{P(X=2)}=\frac{1}{7500}$ and $\frac{P(X=5)}{P(X=3)}=\frac{1}{500}$, then the m 41. $A(2,0), B(0,2), C(-2,0)$ are three points. Let $a, b, c$ be the perpendicular distances from a variable point $P$ on to 42. When the coordinate axes are rotated about the origin through an angle $\frac{\pi}{4}$ in the positive direction, the eq 43. $y-x=0$ is the equation of a side of a $\triangle A B C$. The orthocentre and circumcentre of the $\triangle A B C$ are 44. Two families of lines are given by $a x+b y+c=0$ and $4 a^2+9 b^2-c^2-12 a b=0$. Then, the line common to both the famil 45. Two non-parallel sides of a rhombus are parallel to the lines $x+y-1=0$ and $7 x-y-5=0$. If $(1,3)$ is the centre of the 46. If the equations $3 x^2+2 h x y-3 y^2=0$ and $3 x^2+2 h x y-3 y^2+2 x-4 y+c=0$ represent the four sides of a square, the 47. The radius of the circle having three chords along Y-axis, the line $y=x$ and the line $2 x+3 y=10$ 48. Among the chords of the circle $x^2+y^2=75$, the number of chords having their mid-points on the line $x=8$ and having t 49. The equation of the circle which touches the circle $S \equiv x^2+y^2-10 x-4 y+19=0$ at the point $(2,3)$ internally and 50. If $P\left(\frac{7}{5}, \frac{6}{5}\right)$ is the inverse point of $A(1,2)$ with respect to a circle with centre $C(2,0 51. If the circle $S=0$ intersect the three circle $$ \begin{aligned} & S_1 \equiv x^2+y^2+4 x-7=0 \\ & S_2 \equiv x^2+y^2+y 52. If a tangent of the circle $x^2+y^2+2 x+2 y+1=0$ is radical axis of the circles $x^2+y^2+2 g x+2 f y+c=0$ and $2 x^2+2 y 53. If the angle between the tangents drawn to the parabola $y^2=4 x$ from the points on the line $4 x-y=0$ is $\frac{\pi}{3 54. The normal at a point on the parabola $y^2=4 x$ passes through a point $P$. Two more normals to this parabola also pass 55. The circumcenter of the equilateral triangle having the three points $\theta_1, \theta_2, \theta_3$ lying on the ellipse 56. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(b>a)$ is an ellipse with eccentricity $\frac{1}{\sqrt{2}}$. If the angle of intersec 57. The number of common tangents that can drawn to the curves $\frac{x^2}{16}-\frac{y^2}{9}=1$ and $x^2+y^2=16$ is 58. Let $A(\alpha, 4,7)$ and $B(3, \beta, 8)$ be two points in space. If $Y Z$ plane and $Z X$-plane respectively divide the 59. The direction ratios of the line bisecting the angle between the $X$-axis and the line having direction ratios $(3,-1,5) 60. If the plane $-4 x-2 y+2 z+\alpha=0$ is at a distance of two units from the plane $2 x+y-z+1=0$, then the product of all 61. $$ \mathop {\lim }\limits_{x \to 0} \frac{\sqrt{\cos x}-\sqrt[3]{\cos x}}{\sin ^2 x}= $$ 62. Let $f:[-1,2] \rightarrow R$ be defined by $f(x)=\left[x^2-3\right]$ where $[$. denotes greatest integer function, then 63. If $f(x)=\left\{\begin{array}{cc}x^2\left|\cos \frac{\pi}{2}\right|, & x \neq 0 \\ 0, & x=0\end{array}\right.$, then at 64. The set of all values of $x$ for which $f(x)=\| x|-1|$ is differentiable is 65. $$ \begin{aligned} &\text { If } y=f(x)^{g(x)} \text { and } \frac{d y}{d x}=y\left[H(x) f^{\prime}(x)+G(x) g^{\prime}(x 66. If $x=t-\sin t, y=1-\cos t$ and $\frac{d^2 y}{d x^2}=-1$ at $t=k, k>0$ then $\lim _{i \rightarrow K} \frac{y}{x}=$ 67. For the curve $\frac{x^n}{a^n}+\frac{y^n}{b^n}=2,(n \in N$ and $n>1)$ the line $\frac{x}{a}+\frac{y}{b}=2$ is 68. The height of a cone with semi-vertical angle $\frac{\pi}{3}$ is increasing at the rate of 2 units $/ \mathrm{min}$. The 69. The function $f(x)=2 x^3-9 a x^2+12 a^2 x+1$ where $a>0$ attains its local maximum and local minimum at $p$ and $q$ resp 70. Consider all functions given in List I in the interval [1,3]. The list II has the value of ' $c$ ' obtained by applying 71. If the percentage error in the radius of a circle is 3 , then the percentage error in its area is 72. $$ \begin{aligned} I_1 & =\int \frac{e^x}{e^{4 x}+e^{2 x}+1} d x, I_2 \\ & =\int \frac{e^{-x}}{e^{-4 x}+e^{-2 x}+1} d x, 73. If $\int \frac{1}{x} \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}} d x=2 f(x)-2 \sin ^{-1} \sqrt{x}+c$, then $f(x)=$ 74. $$ \begin{aligned} & \int \frac{3 x+2}{4 x^2+4 x+5} d x=A \log \\ & \left(4 x^2+4 x+5\right)+B \tan ^{-1}\left(\frac{2 x 75. Consider the following Assertion $$ \begin{aligned} & \text { (A) } \begin{array}{r} \sqrt{x-3}\left(\sin ^{-1}(\log x)+ 76. $$ \lim _{n \rightarrow \infty} \frac{\left(2 n(2 n-1) \ldots .(n+2)(n+1)^{1 / n}\right.}{n}= $$ 77. The area of the region bounded by $y=x^3, X$-axis, $x=-2$ and $x=4$ is 78. If $\int_0^{\frac{\pi}{2}} \tan ^{14}\left(\frac{x}{2}\right) d x=2\left[\sum_{n=1}^7 f(n)-\frac{\pi}{4}\right]$, then $ 79. The differential equation of the family of all circles of radius ' $a$ ' is 80. If the general solution of $\left(1+y^2\right) d x=\left(\tan ^{-1} y-x\right) d y$ is $x=f(y)+c e^{-\tan ^{-1} y}$, the

Physics

1. The force of mutual attraction between any two objects by virtue of their masses is 2. The error in the measurement of force acting normally on a square plate is $3 \%$. If the error in the measurement of th 3. For a particle moving along a straight line path, the displacements in third and fifth seconds of its motion are 10 m an 4. The vertical displacement ( $y$ in metre) of a projectile in term of its horizontal displacement ( $x$ in metre) is give 5. A block of mass $\sqrt{2} \mathrm{~kg}$ is placed on a rough horizontal surface. A force ' $F$ ' acting upwards at an an 6. If the kinetic energy of a body moving with a velocity of $(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}) \m 7. A body of mass 0.5 kg is supplied with a power ' $P$ ' (in watt) which varies with time ' $f$ ' (in second) as $P=3 t^2+ 8. A solid sphere of mass 2 kg and radius 0.5 m is rolling without slipping on a horizontal surface. The ratio of the rotat 9. If the length of a thin uniform rod is ' $L$ ' and the radius of gyration of the rod about an axis perpendicular to its 10. The force ( $F$ in newton) acting on a particle of mass 90 g executing simple harmonic motion is given by $F+0.04 \pi^2 11. Which of the following is incorrect about the gravitational force between two bodies? 12. A steel rod with a circular cross-section of diameter 1 cm and another steel rod with a square cross-section of side 1 c 13. A cube of side 40 cm is floating with $\frac{1}{4}$ th of its volume immersed in water. When a circular disc is placed o 14. The maximum length of water column that can stay without falling in a vertically held capillary tube of diameter 1 mm an 15. A steel pendulum clock manufactured at $32^{\circ} \mathrm{C}$ and working at $47^{\circ} \mathrm{C}$ is nearly (Coeffic 16. A metal metre scale that is accurate up to 0.5 mm is made at a temperature of $25^{\circ} \mathrm{C}$. The range of temp 17. A Carnot engine uses diatomic gas as a working substance. During the adiabatic expansion part of the cycle, if the volum 18. The ratio of the average translational kinetic energies of hydrogen and oxygen at the same temperature is 19. The air columns in two tubes closed at one end vibrating in their fundamental modes produce 2 beats per second. The numb 20. A car moving towards a cliff emits sound of frequency ' $n$ '. If the difference in frequencies of the horn and its echo 21. A straight metal rod of length 6 cm is placed along the principal axis of a concave mirror of focal length 9 cm such tha 22. A ray of light incidents at an angle of $9.3^{\circ}$ on one face of a small angle prism of refracting angle $6^{\circ}$ 23. The distance for which ray optics becomes a good approximation for an aperture of 0.3 cm and a light of wavelength $6000 24. The electrostatic force between two charges kept in air is $F$. If $30 \%$ of the space between the charges is filled wi 25. 729 small identical spheres each charged to an electric potential 3V combine to form a bigger sphere. The electric poten 26. For the circuit shown in the figure, the current through $6 \Omega$ resistor connected between the junctions $A$ and $B$ 27. The area of cross-section of a potentiometer wire is $6 \times 10^{-7} \mathrm{~m}^2$. The potential difference per unit 28. As shown in the figure, a uniform straight wire of length $30 \sqrt{3} \mathrm{~cm}$ is bent in the form of an equilater 29. A proton moving with a velocity of $8 \times 10^5 \mathrm{~ms}^{-1}$ enters a uniform magnetic fleld normal to the direc 30. At a certain place in the magnetic meridian the Earth's magnetic field is twice its vertical component. The ratio of hor 31. A coil of resistance $16 \Omega$ is placed with its plane perpendicular to a uniform magnetic field whose flux ( $\phi$ 32. The small energy losses in transformers due to eddy currents can be reduced by 33. If the electric field of a plane electromagnetic wave is $E_z=60 \sin \left(0.5 \times 10^3 x+1.5 \times 10^{11} t\right 34. In a photoelectric experiment, the slope of the graph drawn between stopping potential along $Y$-axis and frequency of i 35. The maximum wavelength of incident radiation required to ionize a hydrogen atom in its ground state is nearly 36. When an element ${ }_{90}^{232} \mathrm{Th}$ decays into ${ }_{82}^{208} \mathrm{~Pb}$, the number of $\alpha$ and $\bet 37. During the disintegration of a radioactive nucleus of mass number 208 at rest, two alpha particles each with kinetic ene 38. The current amplification factor of a transistor in common emitter configuration is 80 . If the emitter current is 2.43 39. The negative feedback in an amplifier 40. If the frequencies of the carrier wave and message signal are 1 MHz and 28 kHz respectively, then the frequencies of the

TG EAPCET 2025 (Online) 4th May Evening Shift

MCQ (Single Correct Answer)

-0

Consider all functions given in List I in the interval [1,3]. The list II has the value of ' $c$ ' obtained by applying Lagrange's mean value theorem on the function of List I . Match the function and values of ' c '

$$ \begin{array}{llll} \hline & \text { List I } & & \text { List II } \\ \hline \text { A } & |x-1| & \text { I } & 2 \log \left(e^3+e^2\right) \\ \hline \text { B } & \log x & \text { II } & 2 \\ \hline \text { C } & x^2+x+1 & \text { III } & \log _3 e^2 \\ \hline \text { D } & e^x & \text { IV } & \sqrt{2} \\ \hline & & \text { V } & \log \left(\frac{e^3-e}{2}\right) \\ \hline \end{array} $$

A-II, B-V, C-IV, D-III

A-II, B-I, C-IV, D-III

A-IV, B-V, C-II, D-I

A-IV, B-III, C-II, D-V

TG EAPCET 2025 (Online) 4th May Evening Shift

MCQ (Single Correct Answer)

-0

If the percentage error in the radius of a circle is 3 , then the percentage error in its area is

$\frac{3}{2}$

TG EAPCET 2025 (Online) 4th May Evening Shift

MCQ (Single Correct Answer)

-0

$$ \begin{aligned} I_1 & =\int \frac{e^x}{e^{4 x}+e^{2 x}+1} d x, I_2 \\ & =\int \frac{e^{-x}}{e^{-4 x}+e^{-2 x}+1} d x, \text { then } I_2-I_1= \end{aligned} $$

$\frac{1}{2} \log \left(\frac{e^{2 x}-e^{-2 x}+1}{e^{2 x}+e^{-2 x}-1}\right)+C$

$\frac{1}{2} \log \left(\frac{e^{2 x}-e^{-2 x}-1}{e^{2 x}+e^{-2 x}+1}\right)+C$

$\frac{1}{2} \log \left(\frac{e^{2 x}+e^{-x}+1}{e^{2 x}+e^{-x}-1}\right)+C$

$\frac{1}{2} \log \left(\frac{e^x+e^{-x}-1}{e^x+e^{-x}+1}\right)+C$

TG EAPCET 2025 (Online) 4th May Evening Shift

MCQ (Single Correct Answer)

-0

If $\int \frac{1}{x} \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}} d x=2 f(x)-2 \sin ^{-1} \sqrt{x}+c$, then $f(x)=$

$\operatorname{sech}^{-1} \sqrt{x}$

$\operatorname{cosec}^{-1} \sqrt{x}$

$\log \left(\frac{1+x}{\sqrt{x}}\right)$

$\log \left(\frac{\sqrt{1+x}-1}{\sqrt{x}}\right)$

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